CatDat

Implication Details

Claim: If a category has a natural numbers object and has a strict terminal object, then it is one-way.

Proof: By assumption, $z : 1 \to N$ is an isomorphism. Therefore, the terminal object $1$ is a NNO with $z = \id_1$ and $s = \id_1$. This precisely means that for all $f : A \to X$ and $g : X \to X$ there is a unique $\Phi : A \to X$ with $\Phi = f$ and $\Phi = g \circ \Phi$. In other words, we have $f = g \circ f$, and therefore $g = \id_X$ (take $f = \id_X$), which proves the claim. (From here one can further deduce that the category is thin.)